Statistical topological data analysis using persistence landscapes

TL;DR

This paper introduces the persistence landscape, a new topological data analysis summary that is compatible with statistical and machine learning tools, and establishes its theoretical properties and applications.

Contribution

The paper proposes the persistence landscape as a novel topological summary that is easy to integrate with statistical methods and proves its stability and inferential properties.

Findings

Persistence landscapes obey a strong law of large numbers.

They satisfy a central limit theorem in a Banach space.

The method provides lower bounds for bottleneck and Wasserstein distances.

Abstract

We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.